Robot motion planning, weights of cohomology classes, and cohomology operations
Robot motion planning, weights of cohomology classes, and cohomology operations
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DOI：
10.1090/s0002-9939-08-09529-4
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发表时间：
2007-06
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影响因子：
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通讯作者：
M. Farber;Mark Grant
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文献类型：
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作者：
M. Farber;Mark Grant
解决运动规划问题的算法的复杂性是通过系统配置空间 X 的同伦不变量 TC(X) 来衡量的。先前已知的 TC(X) 下界使用 X 的上同调代数结构。在本文中，我们展示了如何使用上同调运算来锐化 TC(X) 的这些下界。作为该技术的应用，我们明确地计算了各种透镜空间的拓扑复杂性。该论文的结果受到 E. Fadell 和 S. Husseini 关于 Lusternik - Schnirelmann 范畴经典下界中出现的上同调类权重的研究的启发。在本文的附录中，我们给出了其结果的广义版本的非常简短的证明。
The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations can be used to sharpen these lower bounds for TC(X). As an application of this technique we calculate explicitly the topological complexity of various lens spaces. The results of the paper were inspired by the work of E. Fadell and S. Husseini on weights of cohomology classes appearing in the classical lower bounds for the Lusternik - Schnirelmann category. In the appendix to this paper we give a very short proof of a generalized version of their result.